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Although the FMR linewidth is narrow at room temperature (corresponding to a damping coefficient $\\alpha$ = (9.0 $\\pm$ 0.2) $\\times 10^{-4}$), comparable to previous results for high-quality YIG films of similar thickness, the linewidth increases strongly at low temperatures, by a factor of almost 30. This increase cannot be explained as due to two-magnon scattering from defects at the sample interfaces. 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